Lösungen/Solutions/E1S1AlUe/Ex11

L=Lektion/Leçon-- Sprechstunde/heures de consultation
W=Work/Arbeit/Travail
A  =Anhang/  Annexe / Appendice

Ueb/Ex 1      W

rg1[t_] := {-2, 4} + t {7, 2};
rg2[t_] := {8, 3} + t {-4, 5};

s1a = 
      Solve[rg1[λ] == rg2[μ], {λ, μ}] // 
        Flatten

s1a // N

res1a = {rg1[λ], rg2[μ]} /. s1a

res1a // N

ParametricPlot[Evaluate[rg1[t]], {t, -4, 4}];

ParametricPlot[
        Evaluate[{rg1[t], rg2[t]}], {t, -4, 4}];

h[t_] := 
      rg1[t] + {-2, 7}/Sqrt[(-2)^2  +  7^2] 4

h[t] // N

ParametricPlot[
        Evaluate[{rg1[t], rg2[t], h[t]}], {t, -4, 4}];

s1b = 
      Solve[h[λ] == rg2[μ], {λ, μ}] // 
        Flatten

s1b // N

{h[λ], rg2[μ]} /. s1b  // N

Ueb/Ex 2      W

a

g[t_] := {-2, 4, 1} +  t {7, 2, -1}

Φ[λ_, μ_] := {1, 
          0, -2} + λ {4, 1, 4} + μ {-3, -2, 
            1}

s2a = 
      Solve[g[t] == Φ[λ, μ], {t, λ, μ}] // 
        Flatten

c

res2a // N

b

Ψ[λ_, μ_] := {0, 1, 
          1} + λ {-1, 2, 3} + μ {1, -1, 1}

s2b = 
      Solve[Ψ[λ, μ] == Φ[ζ, η], {μ, ζ, η}] // Flatten

res2b = {Ψ[λ, μ], Φ[ζ, η]} /. s2b

res2b // N

gs[λ_] := res2b[[1]];  gs[λ]

ParametricPlot3D[Evaluate[gs[λ]], {λ, -1.8, 0}, 
        ViewPoint -> {2.265,  1.593,  1.945}];

Solve[gs[λ] == {x, y, 0}, {x, y, λ}]  // 
      Flatten

% // N

Solve[gs[λ] == {x, 0, z}, {x, z, λ}]  // 
      Flatten

% // N

Solve[gs[λ] == {0, y, z}, {y, z, λ}]  // 
      Flatten

% // N

Ueb/Ex 3      L

a

r = Sqrt[1^2 + 2^2 + 3^2]

α = ArcCos[1/r]

N[α]

β = ArcCos[2/r]

N[β]

γ = ArcCos[3/r]

N[γ]

b

x = 1; y = 2; z = 3;

aStrich = {x  Cos[φ] - y  Sin[φ], 
        x  Sin[φ] + y  Cos[φ], z}

φ = π/7;  aStrich

% // N

Ueb/Ex 4      W

a

Remove[x, y, z]

Φ[λ_, μ_] := {0, 1, 
          1} + λ {-1, 2, 3} + μ {1, -1, 1}

s4 = 
      Solve[Φ[λ, μ] == {x, y, 
              z}, {λ, μ, x}]  // Flatten

x1 = x /. s4

equat = (x - x1 == 0 // Expand)

equat[[1]] 5/3 // Expand

Remove[h]

h[{x_, y_, z_}] := (-1 + 5  x/3 + 4  y/3 - z/3) 
        5/3 1/Sqrt[(5/3)^2 + (4/3)^2 + (-1/3)^2]

?h

h[{0, 1, 1}]

h[Φ[λ, μ]] // Simplify

d = h[{15, 20, 18}]

d // N

b

gH[t_] := {15, 20, 18} +  t {5/3, 4/3, -1/3}

s4b = 
      Solve[gH[t] == Φ[λ, μ], {t, λ, μ}] // 
        Flatten

res4b = {gH[t], Φ[λ, μ]} /. s4b

res4b // N